Bayesian Analysis in Regression Models Using Pseudo-Likelihoods

: Certain best population problems adopting a Bayesian approach were investigated. One main advantage of such an approach is that satisfactory decision procedures can be obtained in the presence of nuisance parameters. The use of Bayes' theorem allows one to analyze best population problems from many points of view. A general description of best population problems is outlined. The criterion for 'bestness' is regarded as a 'utility' function of the statistician. The decision procedure then adopted is based upon the principle of maximizing posterior expected utility. The application of this procedure when sampling from normal populations and exponential populations was discussed. The criterion defining the best population is taken to be the coverage of the population considered in a certain given interval. The procedures are shown to be consistent. The decision analysis by considering the pos terior distribution of the criterion is presented. This extension enables other decision procedures to be proposed which may be more appropriate in certain situations than that resulting from the principle of maximizing posterior expected utility.

Binary-discrimination tasks useful to arms-control-treaty verification were performed by applying mathematical observer models used by the medical-imaging community. It is a difficult task, as the monitor needs to verify a measured object is a warhead while the host wants to prevent dissemination of sensitive information on their weapons. Inspection objects were classified by their projection data, without reconstructing an image, which often contains sensitive information. Furthermore, the models process data event-by-event, with only a scalar test statistic being updated before the observed data is purged from memory, preventing the aggregation of information which would be sensitive. Template matching models were developed from a set of calibration data on these objects, with the ultimate goal being to develop an observer model that stores only non-sensitive information sufficient for confirmation. These models were also analyzed in the presence of nuisance parameters — unknowns in the objects or imaging system that affect the projection data but aren't of interest to the discrimination task. Examples of these include object location and orientation. The Hotelling observer and channelized Hotelling observer were modeled and their benefits to information security analyzed. The Hotelling observer-the ideal linear observer when the statistics of the data are Gaussian-stores only a set of weights which are the product of the inverse average covariance matrix and difference in mean data between the two objects in the discrimination task. When testing a source, an inner product of the Hotelling weights and binned testing data is taken, resulting in a scalar that is thresholded to make a decision. If nuisance parameters are present, the mean and covariance matrix are found by averaging not only over the always present Poisson noise, but the nuisance parameter distributions as well. Hence, even if detector data is gathered from multiple realizations of an object, the Hotelling weights will be a single smeared out data set the size of the measured data. The Hotelling weights also sifts out all information other than the differences between the two objects. Hence, if the monitor was to gain access to the weights and apply the inverse to their test statistic, they could only back out a scaled version of the template, not the image. The channelized Hotelling observer-a common tool used in medical image quality assessment-was also investigated. This method drastically reduces the size of the data by applying a channeiizing matrix to the binned testing data set. An optimal set of weights for the channelized data can then be found, and the inner product between these weights and the channelized vector results in the scalar test statistic. Optimal performance is retained by optimizing the matrix to maximize the SNR2 between the test-statistic distributions for the two sources in the task. The channelized Hotelling observer gives the monitor access to multiple non-sensitive test statistics. In practice, the channelizing matrix could be implemented in hardware or software behind an information barrier. The addition of penalty terms to the channelized Hotelling observer offers further potential for this method. Individual channel performance could be penalized, leading to a large number of non-sensitive channels that the monitor can use to verify the channelization routine is working as described without gaining access to sensitive information. Noise could be added to the resulting channels, causing additional reduction of the total stored information. Finally, if the host can define what information in its object geometries is sensitive in advance, they could optimize the differences between the two objects in the task while penalizing out the sensitive information, creating a non-sensitive channeiizing matrix that could be shared with the monitor. To test these models, Monte Carlo simulations were performed with the GEANT4 toolkit. Photons were tracked from plutonium inspection objects developed by Idaho National Laboratory. We simulated the Fast-Neutron Imaging system designed by Oak Ridge National Laboratory and Sandia National Laboratories, which consists of 40×40 1cm2 liquid scintillator pixels with a plastic coded aperture. Observer models were evaluated using the area under the ROC curve. This work is supported by the Office of Defense Nuclear Nonproliferation Research and Development, Nuclear Weapon and Material Security Team. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. (SAND2015-10391A).

The Poisson regression model is widely used for count data. This model assumes equidispersion. In practice, equidispersion is seldom reflected in data. However, in real-life data, the variance usually exceeds the mean. This situation is known as overdispersion. Negative binomial distribution and other Poisson mix models are often used to model overdispersion count data. Another extension of the negative binomial distribution in another model for count data is the univariate generalized Waring. In addition, the model developed by Famoye can be used in the analysis of count data. When the count data contains a large number of zeros, it is necessary to use zero-inflated models. In this study, different generalized regression models are emphasized for the analysis of excessive zeros count data. For this purpose, a real data set was analysed with the generalized Poisson model, generalized negative binomial model, generalized negative binomial Famoye, generalized Waring model, and the foregoing zero-inflated models. Log-likelihood, Akaike information criterion, Bayes information criterion, Vuong statistics were used for model comparisons.

The Fisher information about parameters of interest (P‐information) is invariant with respect to nuisance parameters, and induces an information inequality associated with likelihood factorization. This information inequality provides a natural basis for measuring information loss due to using only a sublikelihood function for inference. In contrast with the global reparametrization of some previous concepts in the literature, the concepts of P‐ancillarity and P‐sufficiency proposed in this article are characterized by the notion of no pointwise information loss with respect to the parameters of interest. A conditional version of P‐sufficiency is also proposed. The asymptotic efficiency of likelihood inference under P‐ancillarity or P‐sufficiency is outlined.

Distributional properties for the generalized p-value for the Behrens–Fisher problem

The problem of comparing the frequentist evidence and the Bayesian evidence in the one‐sided testing problems has been widely treated and many researches revealed that these two methods can reach an agreement approximately. However, most of the previous work dealt mainly with situations without nuisance parameters. Since the presence of nuisance parameters is very common in practice, whether these two kinds of evidence still reach an agreement is a problem worthy of study. In this article, we establish in a systematic way under the exponential distributions the agreement of the Bayesian evidence and the generalized frequentist evidence (the generalizedP‐value) for a variety of one‐sided testing problems where the nuisance parameters are involved.

Supremum score test statistics are often used to evaluate hypotheses with unidentifiable nuisance parameters under the null hypothesis. Although these statistics provide an attractive framework to address non‐identifiability under the null hypothesis, little attention has been paid to their distributional properties in small to moderate sample size settings. In situations where there are identifiable nuisance parameters under the null hypothesis, these statistics may behave erratically in realistic samples as a result of a non‐negligible bias induced by substituting these nuisance parameters by their estimates under the null hypothesis. In this paper, we propose an adjustment to the supremum score statistics by subtracting the expected bias from the score processes and show that this adjustment does not alter the limiting null distribution of the supremum score statistics. Using a simple example from the class of zero‐inflated regression models for count data, we show empirically and theoretically that the adjusted tests are superior in terms of size and power. The practical utility of this methodology is illustrated using count data in HIV research.

This chapter introduces statistical methods for twin and sibling studies using random effects models. The classic twin study begins from assessing the variance of a trait (called a phenotype by geneticists) in a large group, and attempts to estimate how much of this is due to genetic effects (heritability), how much appears to be due to shared environmental effects, and how much is due to unique environmental effects — events occurring to one twin but not another. This chapter shows how three families of random effect models, random effects models for normally distributed traits, generalized linear mixed models for binary and count data, and frailty models for survival data may be used for modelling variances in longitudinal, or life course setting. The methods are illustrated using data from The Swedish Adoption/Twin Study of Aging. The appendix provides the R and Winbugs codes that are used in the analyses.

The purpose of this chapter is to present recent advances in models for genomic-enabled prediction developed for ordinal categorical and count data. For both models we provide details of their corresponding derivation and then apply them to a real data set. The proposed models were derived using a Bayesian framework. Bayesian logistic ordinal regression (BLOR) and Bayesian negative binomial regression (BNBR) make use of the Polya-Gamma distribution to produce an analytic Gibbs, a sampler with similar full conditional distributions of a model with Gaussian response and can be used for complex data sets as those that arise in the context of genomic selection where the sample size usually is smaller than the number of covariates (markers). We illustrate the proposed models using simulation and a real data set. Results indicate that our models for ordinal categorical and count data are a good alternative for analyzing ordinal and count data in the context of genomic-enabled prediction.

A broad class of generalized linear mixed models, e.g. variance components models for binary data, percentages or count data, will be introduced by incorporating additional random effects into the linear predictor of a generalized linear model structure. Parameters are estimated by a combination of quasi‐likelihood and iterated MINQUE (minimum norm quadratic unbiased estimation), the latter being numerically equivalent to REML (restricted, or residual, maximum likelihood). First, conditional upon the additional random effects, observations on a working variable and weights are derived by quasi‐likelihood, using iteratively re‐weighted least squares. Second, a linear mixed model is fitted to the working variable, employing the weights for the residual error terms, by iterated MINQUE. The latter may be regarded as a least squares procedure applied to squared and product terms of error contrasts derived from the working variable. No full distributional assumptions are needed for estimation. The model may be fitted with standardly available software for weighted regression and REML.

This volume has been inspired by the aim of gathering a number of theoretical papers that, though from different schools of inference—frequentist andBayesian, share the same interest towards studying complex statistical models. Indeed, both frequentist and Bayesian inference can entail various drawbacks, due to the complexity or misspecification of the model, or to the presence of many nuisance parameters. These difficulties may be overcome from a theoretical point of view through approximate likelihoods and scoring rules, and from a computational point of view through higher-order asymptotic expansions or computational tools from Bayesian inference. Specifically, the volume focuses on the interplay between Bayesian and frequentist inference in addressing these problems. The goal is to highlight and to disseminate some of the topics emerged in the international Workshop Recent advances in Statistical Inference: Theory and case studies, held in Padova, Italy, March 21–23, 2013. The Workshop has been a forum for the exchange of new ideas and thoughts on recent and ongoing research in both Bayesian methodology and modern likelihood inference. Although the topics covered in this volume are diverse, similar themes recur, as research ismostly fueled by the need to deal with complex models, for which traditional methods do not provide viable solutions. Among the topics presented we have Bayesian/frequentist challenges for categorical data analysis, Bayesian-frequentist sample size determination, integrated likelihood inference in semi parametric regression models, approximate Bayesian computation with modified log-likelihood ratios, theory and applications of proper scoring rules, quasi likelihood approximation of posterior distributions for likelihood-intractable complex models, mitigating multicollinearity with spike-and-slab priors, empirical Bayes methods in classical and Bayesian inference. While not exhaustive, this list should give a feeling of the issues discussed at the Workshop.

This dissertation makes a number of contributions to the unit-level modeling literature. First, the connection between preferential sampling in ecological statistics and informative sampling, which arises in complex surveys for official statistics, is described. Solutions to the problems are reviewed and compared and suggestions for bridging the gap between the two disciplines are made. Second, longitudinal models for Gaussian, binary, and count data in the context of complex surveys are developed and applied to the Household Pulse Survey. Third, a cross-sectional ordinal model that addresses informative sampling is introduced, as well as longitudinal ordinal and nominal models. In addition, efficient variational Bayesian algorithms are presented. Lastly, an extension to wealth data is considered alongside an application to the Survey of Income and Program Participation.

An optimality criterion for vector unbiased statistical estimation functions

SummaryLet (X, A, P)be a given statistical model and letg be the class of all one-to-one, bimeasurable maps g of ( X, A) onto itself such that g is measure-preserving for each P ε P, i.e. Pg−1= P for all P. Let us suppose that there exists a least (minimal) sufficient sub-fieldL. Then, for each LεL, it is true that g−1L is P-equivalent to L for each gεg , i.e., the least sufficient sub-field is almost g -invariant. It is demonstrated that, in many familiar statistical models, the least sufficient sub-field and the sub-field of all almost g -invariant sets are indeed P-equivalent. The problem of data reduction in the presence of nuisance parameters has been discussed very briefly. It is shown that in many situations the principle of invariance is strong enough to lead us to the standard reductions. For instance, given n independent observations on a normal variable with unknown mean (the nuisance parameter) and unknown variance, it is shown how the principle of invariance alone can reduce the data to the sample variance.

Hypothesis tests for a vector-valued parameter are considered in this chapter. A single component of the parameter vector is of interest in the test, and the rest of the components the parameter vector are the “nuisance parameters.” The goal is to find a test that maximizes the power over all the components of the parameter vector. Because of the uniform nature of this optimality, it is possible only under rather restrictive conditions. Thus the Uniformly Most Powerful Unbiased tests are considered.